Why the second partial is not the octave
An ideal string — perfectly flexible, all mass and no stiffness — puts its partials at exact whole multiples of the fundamental. Real piano wire resists being bent, and that resistance adds a restoring force that grows with how sharply the string has to curve. High partials curve more, so they are stiffened more, and the whole series drifts upward. The standard description of that drift needs one number, the coefficient B, and the frequency of the nth partial becomes n times the fundamental times the square root of one plus B times n squared.
With the values sitting in the form — A4 at 440, a 17 inch speaking length, 0.0375 in wire, 29 million psi — the tension comes out near 181 lbf and B near 0.00053. The second partial is then about 1.8 cents above 880 Hz. Tune A5 to stop that partial beating and A5 sits 1.8 cents above 880, not at it. Do that all the way up and the top of the instrument ends up perceptibly sharp of a mathematical keyboard, which is exactly what a piano sounds like and exactly what an electronic keyboard tuned to pure ratios does not.
The fourth power is the whole story of small pianos
Hold the pitch and the wire fixed. Tension follows the square of the speaking length, because pitch depends on length and tension together. Push that through the expression for B and the diameter squared survives on top and the length appears to the fourth power underneath. B goes as d squared over L to the fourth.
That exponent is unforgiving. Take half an inch off 17 and you have shortened the string by 2.9 percent; B rises by 12.7 percent. To halve B you need the string 18.9 percent longer, which on a 17 inch string is another 3.2 inches. Nobody finds three inches in a case that is already built. It is the reason the tenor of a short upright sounds the way it does, and it is not a manufacturing defect, it is geometry.
Thinner wire helps B and costs tension, since tension follows the square of the diameter as well. Scale design is that trade run note by note across eighty-eight of them, and the numbers on this page are one note of it.
Reading the tension figure honestly
The tension line comes from the Mersenne relation and nothing else: frequency is one over twice the length, times the square root of tension over mass per unit length. Rearranged, tension is the mass per unit length times the square of twice the length times the frequency. Every term is either measured or supplied. There is no fudge factor and no hidden constant beyond the conversion from weight per inch to mass.
What that means practically is that the tension figure is only as good as the diameter. Tension follows the diameter squared, so a wire you assumed was 0.0375 and which is really 0.0380 carries 2.7 percent more tension than the page says. A micrometer settles it in five seconds and a gauge number does not, because gauge numbering is a supplier convention rather than a standard and two suppliers can disagree about the same number.
The stress line divides tension by cross section. If a tensile strength is entered from the supplier table, the page prints the ratio and stops there. It does not say whether that ratio is fine, and it will not, because the strength of a specific coil, the condition of the wire after years in a frame, the radius it turns at the bridge pin and the state of the termination are all outside anything a form can see.
Calculated B against measured B
Devices that read individual partials will give you B for a string directly, and it is worth doing on any instrument you are going to spend a day on. Measured and calculated values routinely differ. The model assumes the wire is uniform and that both ends are simply supported, and neither is quite true — a bridge pin termination is somewhere between pinned and clamped, and where it sits on that scale changes the effective stiffness. There is also the front duplex and whatever the capo bar is doing.
When the two disagree, the measurement wins. The calculation is for the case where there is no instrument in front of you: designing a scale, choosing between two wire sizes for a rescale, or working out before you order whether a length you can actually fit will behave.
Questions people ask
What is a normal value for B on a piano string?
There is no normal value this page will assert, because it depends entirely on the scale and on which note you are asking about. What is worth knowing is the shape: B is smallest in the middle of a well-scaled instrument and rises toward both ends, sharply in the top octaves where strings are short and thick relative to their length, and awkwardly in the bass of a short piano where the design ran out of room. Measure a few notes on the instrument you are working on and you will have a curve that means something, which is more use than a number someone else measured on a different piano.
Does inharmonicity mean piano octaves are out of tune?
It means a pure 2 to 1 octave and a beatless octave are two different things on a piano, and only one of them is available. If you set the upper note to exactly double the frequency, it beats against the second partial of the lower note, and that beat is audible. If you set it to stop beating, the frequency ratio is more than 2 to 1 by the number this page prints. Tuners choose the second one, which is why piano tuning is described as stretched. It is not a compromise forced by bad strings; it is what the strings actually do.
Can I use this for a wound bass string?
No, and the page says so in the results. A wound string carries a steel core with a copper wrap over it. The mass per unit length comes from both the core and the wrap, while the bending stiffness comes almost entirely from the core, which is much thinner than the outside diameter. Feeding the outside diameter into an arithmetic that assumes one solid wire gives you a stiffness that is far too high and a tension that is wrong in the other direction. Wound strings are worked from the core diameter and the wrap separately, or measured on the instrument.
Why does the tension depend on the square of the frequency?
Because the Mersenne relation puts frequency proportional to the square root of tension. Invert it and tension is proportional to frequency squared, holding length and wire constant. That is why dropping a whole step is not a small change: two semitones is a frequency ratio of about 0.891, and squared that is 0.794, so a fifth of the tension leaves. The same square appears on the length, which is why a string half as long at the same pitch on the same wire carries a quarter of the tension.
Which input is the answer most sensitive to?
For the tension, the diameter and the length, both squared. For B, the length by a fourth power and the elastic modulus directly. The modulus is the one people guess at, and guessing at it moves B proportionally — a ten percent error in the modulus is a ten percent error in B and roughly a ten percent error in the octave stretch you calculate from it. That is why it is a field with a named source rather than a constant buried in the code.